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What is the class in $\pi_4(S^3) \simeq \mathbb{Z}_2$ of the map

$$\array{ S^{\mathbb{H}} &\longrightarrow& S^{\left(\mathbb{H}_{im}\right)} \\ q &\mapsto& q v \bar q }$$

where $v \in \mathbb{H}_{im} \setminus \{0\}$ is any non-zero imaginary quaternion?

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An XY-Matrix diagram:

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\draw (0,0) -- (2,0) -- (2,-2);

\end{tikzpicture}

cellular model